Detection Techniques for Fading Multipath ... - Semantic Scholar

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 7, NOVEMBER 1998

Detection Techniques for Fading Multipath Channels with Unresolved Components Florence Danilo, Student Member, IEEE, and Harry Leib, Senior Member, IEEE

Abstract— In this paper we consider noncoherent detection structures for multipath Ricean/Rayleigh fading channels. The multipath components are assumed to be unresolved, with known delays. These delays could have been estimated, for example, by using super-resolution techniques or sounding the channel with a wide-band pulse. We show that the Rayleigh channel optimum receiver (R OPT) consists of an “orthogonalization” (or decorrelation) stage and then it implements an optimum decision rule for a resolved multipath channel. Since the optimum decision rule over Ricean channels is in general too complex for implementation, we propose several suboptimum structures such as the quadratic decorrelation receiver (QDR) and the quadratic receiver (QR). The QDR scheme exploits the decorrelation performed on the input samples. The nonlinear term due to the Ricean specular term is replaced by a quadratic form that is more suitable for implementation. Single-pulse performance of these schemes are studied for commonly used binary modulation formats such as FSK and DPSK. This paper shows that it is possible to have diversity-like gains over Ricean/Rayleigh multipath fading channels with unresolved components even if the channel is not fully tracked. Furthermore, this paper demonstrates the importance of using generalizations of RAKE receivers designed to handle the unresolvability condition. For two-path mixed-mode Ricean/Rayleigh channels, it is shown that improved performance can be obtained by using receivers that know the strength of the Ricean specular term. Index Terms—Decorrelators, detection, fading channels, modulation, multipath channels, quadratic receivers.

I. INTRODUCTION ECENT years have witnessed a growing interest in wireless digital communications for Personal Communication Services (PCS) [1]–[3]. Such systems have to operate in the indoor radio environment that can be characterized as a fading multipath channel [4]. Receiver structures and performance for fading multipath channels are therefore important issues for this application of wireless digital communications. The problem of optimal detection over fading multipath channels is a classic subject in communication theory with solid theoretical foundations. Explicit optimal receiver structures when the multipath delays are known have been introduced by Turin [5] and Price and Green [6]. These well-known structures (RAKE receivers) are based on the path resolvability

R

Manuscript received March 2, 1995; revised February 4, 1998. This work was supported under a Grant from the Natural Science and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Electrical Engineering, McGill University, Telecommunications and Signal Processing Laboratory, Montreal, Que., Canada H3A 2A7 (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0018-9448(98)06748-0.

assumption, asserting that the inter-path delays are larger than the signal autocorrelation time. One of the main attributes of these optimal receivers is the exploitation of the time diversity that is inherent in multipath propagation, yielding diversity gains. The path resolvability assumption is reasonable for spread-spectrum systems of large bandwidth. For narrowband systems, however, path resolvability cannot be ensured in an indoor environment due to the relatively small interpath delays. Therefore, detection techniques for fading multipath channels without the path resolvability condition are of significant interest for PCS. The optimal receiver for multipath Rayleigh channels has been considered by Aiken [7]. Its performance was considered in [8] only for widely orthogonal or uniformly orthogonal signals (signals that are orthogonal to all time shifts). Under these circumstances, the multipath channel is resolved. Mazo [9] has evaluated the matched-filter bound on performance for two-path Rayleigh fading channels assuming the channel to be known exactly. In [9] it was clearly shown that diversitylike improvement at high signal-to-noise ratio (SNR) can be achieved even without path resolvability. Alles and Pasupathy [10], [11] considered receiver structures over two-path Rayleigh for different levels of channel knowledge and found two-fold diversity-like effects in the performance of envelope orthogonal Frequency-Shift Keying (FSK) and variants of chirp or linear frequency sweep modulation. Our work is more general, considering the situation when the channel is not fully tracked, the multipath not resolved, the fading is mixed-mode Ricean/Rayleigh as well as pure Rayleigh with more than two paths, and commonly used modulation schemes. A channel is said to be mixed-mode Ricean/Rayleigh if the first path gain is Ricean-distributed and the other path gains are Rayleigh-distributed. This channel is of practical interest and may correspond to transmission with a line of sight. For example, it belongs to the set of typical channel impulse responses defined in the GSM standard [12]. In this paper we derive noncoherent receiver structures for an -path Ricean/Rayleigh channel when the multipath delays are assumed to be known but unresolved. These multipath delays could have been estimated by using super-resolution techniques [13], or by sounding the channel with a wideband pulse. The multipath component phase shifts are assumed to and since it is in be uniformly distributed between general very difficult to get accurate estimates of the multipath phases [4, p. 953]. Existing channel estimation [15]–[18] and superresolution techniques [13], [14] provide estimates of the interpath delays and amplitudes but not of the carrier phases.

0018–9448/98$10.00  1998 IEEE

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DANILO AND LEIB: FADING MULTIPATH CHANNELS WITH UNRESOLVED COMPONENTS

In this paper, the receiver structures are derived for arbitrary binary modulation schemes; however, the performance is considered for commonly used modulation formats such as variations of Frequency-Shift Keying (FSK) and Differential Phase-Shift Keying (DPSK). Following [9] we determine single-pulse performance (equivalent to matched-filter bounds) but without fully tracking the channel. Instead, only a secondorder statistic of the channel path gains is assumed to be known to the receiver. For small interpath delays, it is to be expected that the effect of intersymbol interference is small, making the single pulse bound close to the actual performance for sequential transmission. The paper is organized as follows. Section II considers receiver structures for Ricean/Rayleigh channels. Section III presents error-rate calculation techniques. Performances of variations of binary FSK and DPSK over two- and three-path Rayleigh fading channels are studied in Section IV. In particular, we will assess the diversity-like gains achievable over such channels when the multipath is unresolved. Section V examines the impact of a Ricean component on the performance by considering a two-path mixed-mode Ricean/Rayleigh channel. The Ricean component can model a line-of-sight path [4]. Finally, Section VI presents the conclusions. II. RECEIVER STRUCTURES A. Channel Modeling Assume transmission of one out of bandpass signals of finite energy over a fading multipath channel. For convenience, possible transmitted signals will be represented by the . Under the hypothesis , the their complex envelope is given by complex envelope of the received signal

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is Gaussian, has a finite mean-square value on the observation and is statistically independent of . Furinterval is much thermore, we assume that the observation interval longer than the multipath delays and the delayed signals

are linearly independent and contained in the observation interval. Let us recall that the resolvability condition requires that all time-shifted versions of the signals are orthogonal, i.e.,

In our analysis, only linear independence is required which is a much weaker constraint compared to orthogonality. It can be easily shown that a square integrable waveform whose Fourier transform is nonzero over at least one interval of nonzero measure, will satisfy the linear independence condition. Furthermore, the linear independence condition is satisfied for any time-limited waveform. We use the following notation: bold capital letters denote matrices and bold lowercase letters denote vectors, , , and denote, respectively, the transposition, complex conjugation, and Hermitian conjugation of a matrix or vector. The th is denoted as and the th entry of entry of a matrix . The diagonal matrix composed a vector is denoted as is denoted by and of the main diagonal entries of the lower triangular matrix composed of the lower triangular with zero main diagonal entries is denoted by elements of . We define

for

(1) where are independent circularly complex Gaussian random , variance variables [19] with mean , and , the phase shifts of the multipath components, are independent, uniformly distributed random and . The multipath delays are variables between if . The effect of the assumed to be known and channel noise is modeled by an additive zero-mean circularly satisfying complex Gaussian process . For Ricean multipath channels each path can be considered as the phasor sum of two components: a Rayleigh component with a uniformly distributed phase and a fixed (specular) component. Furthermore, lack of any phase reference at the receiver is reflected by the uniformly distributed associated with each multipath component random phases , the signal in (1). For all , conditioned on process

where (null set) by convention. Since for all , the energies of the signals are identical and denoted as . We may also define the as correlation matrix of the signal under where

The matrix is the Grammian for the inner product. are linearly Since the signals is positive-definite Hermitian [20, p. 74]. independent, The covariance matrix of the channel is diagonal positive, since definite with th diagonal entry

B. Optimal Decision Rule for an -Path Ricean Channel With multipath Ricean channels, when is fixed, we have the classical problem of detecting a continuous-time Gaussian random signal

in additive white Gaussian noise [21, pp. 419–421]. A minimum probability of error receiver forms the likelihood ratio


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between each one of the hypotheses and a null hypothesis . The decision is made in favor of the largest likelihood ratio [21, p. 11]. The follows from Karhunen–Loéve discrete representation of expansion and exists since the signal is a second-order mean-square-continuous random process [21, pp. 379–380]. under given The covariance of the signal process is

(5f) and

is given by (6) (7)

where (7a) with corresponding mean

(7b)

It is recognized that the covariance of the signal process and it is a finite-dimensional kernel is independent of with well-known eigenvalues and eigenfunctions [22, p. 55]. Notice that in this paper the definition of the eigenvalues is similar to that in [21, p. 379–380]. As shown in Appendix A, the covariance function of the signal process has at most positive eigenvalues and corresponding eigenfunctions. The eigenvalues of the signal process are . Its eigenfunctions the eigenvalues of the matrix are given by (2) where the equations

,

is an

matrix that satisfies (3)

and (4) is the identity matrix and denotes the diagonal matrix . with From (27) in Appendix B, the log-likelihood ratio is given by

(5) where

(5a)

This receiver is illustrated in Fig. 1. The decision vari(7b) can be obtained by using a bank of matched able filters. For example, the following bank of matched filters can be used with a sampling time at : for However, since

elsewhere. for all , for

the decision variable can be obtained by sampling the at (cf. Fig. 1), or by output of the matched filter using a tapped-delay line. The first term in the right side of (5) is a biased quadratic form of the input signal samples. In this paper, a biased quadratic form refers to the sum of a quadratic form and a bias term. The second term in the right side of (5), however, is nonlinear and depends on the multidimensional defined in (5a). Appendix B integral of the function shows how a close-form solution for the integral of can be found for the two-path case in terms of an infinite series of Bessel functions. The technique used in Appendix B can be extended to provide a close-form solution for the integral for the -path case. For the -path Ricean channel, of is to be an -fold path integral need to be solved since integrated over . First integration with respect to is performed yielding Then integration with respect to is done as follows. Similar to in the two-path case, can be expressed as the difference of a complex and a complex term independent term dependent on rotated by . By applying Neumann’s addition of theorem [23, p. 358],

(5b) (5c) (5d) (5e)

is replaced so that its dependence on appears only in cosine terms. Hence integration with respect to can be performed, and the result involves modified Bessel



Fig. 1.

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Optimum receiver for a Ricean multipath channel.

functions of integer orders. Integration with respect to is quite similar to the previous integration. All Bessel functions should be expressed as the arguments that depend on and a difference of a complex term dependent on rotated by . Then complex term independent of Graf’s generalization of Neumann’s theorem [23, p. 361] should be applied individually to all Bessel functions that . The dependence on appears then in depend on can cosine or sine terms and integration with respect to be performed. By performing similar steps for integration over , and so on, it can be shown that the close-form solution for the general case involves multidimensional infinite series of products of Bessel functions. Note that although the phase appears in (5c), the optimal receiver does not require of the knowledge of the phase of the specular term. In fact, cancellation of the phase occurs after the integration of (see example of likelihood for two-path channels in Appendix B). Notice that the optimal receiver reduces to that derived by Turin in [5] when the multipath is assumed to be resolved. is independent of the hypothesis In that special case, and equals , is the identity matrix, and is and equals . From (5), it is seen independent of that the likelihood ratio associated with the general Ricean since . channel involves indirectly a matrix is maintained when the Furthermore, this dependence on specular components vanish (Rayleigh channels). Therefore, the limiting case of Rayleigh channels will provide much and, more generally, insight on the purpose of the matrix

on the operations performed by the optimum receiver for Ricean channels. Rayleigh channels are of practical interest since they may represent transmission without a line of sight. C. Optimal Receiver for Multipath Rayleigh Channels (R OPT) For a Rayleigh multipath channel, . Recall that under the th hypothesis, the received signal is given by (1). For Rayleigh fading channels, the paths magnitude (i.e., the magnitude of the complex Gaussian random variables) are Rayleigh-distributed. The carrier phase shifts are independent, uniformly distributed and absorb the variables . Therefore, an equivalent model for such a channel is given by

The log-likelihood ratio (5) reduces to

(8) or to the equivalent form


(9)

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The first form of the log-likelihood (8) is essentially the receiver found by Aiken in [7], assuming a zero Doppler path shift. The second form (9) provides an important insight to the operations performed by the optimal receiver. In Section II-B we performed a linear transformation on the signals to obtain an orthogonal basis , (2). Therefore, the th hypothesis can be equivalently expressed (since the linear transformation is invertible) as

where and . Under each are independent, hypothesis the new random variables as shown in circularly complex Gaussian with covariance the following:

Similarly, from (5e), (7a), and (3) we have and thus from (5b)

Therefore, we can see that at low , for the Ricean channel optimal receiver the decorrelation operation on the input signal vanishes. This is to be expected since a side effect of the decorrelation operation is to enhance the white background is still used in channel noise. The decorrelation matrix . the bias term The next section will focus on one special channel of practical importance, the mixed-mode Ricean/Rayleigh channel. This channel could represent transmission with a line of sight. D. Receivers Structures for Mixed-Mode Ricean/Rayleigh 1) Optimal Receiver: A multipath fading channel is said to be mixed-mode Ricean/Rayleigh if the first path gain is Ricean-distributed and the other path gains are Rayleigh-dis. For a mixed-mode tributed, i.e., Ricean/Rayleigh channel, (5a) reduces to

Therefore, the log-likelihood ratio (5) reduces to where

if Therefore, under each hypothesis, the received signal can be represented as a linear combination of orthogonal functions weighted by uncorrelated Gaussian random variables, similar to the resolvable multipath case. The log-likelihood ratio for a resolved multipath Rayleigh fading channel is given by [24]

Assume that is small, then is small and using the first terms of the Taylor’s series expansion of the modified Bessel function of the first kind, we have

(10) From (9) and (10) we see that the Rayleigh channel optimal receiver for unresolved multipath channels consists of an orthogonalization (or decorrelation) stage and then implements an optimal decision rule for a resolved multipath channel with transformed signals. To assess the improvement due to the decorrelation stage, we will consider a receiver similar to R OPT except that it does not perform decorrelation. This is the quadratic receiver for Rayleigh fading channels (R QR) whose decision variable is based on (10). The decorrelation is also employed in the more general Ricean multipath optimal receiver. From (5) we can see that the optimal receiver performs a decorrelation on the input samples as well as nonlinear operations related to the Ricean , from (5e) and (3) specular term. Note that for low regardless of ,

thus from (7)

(11) is large, then is large and using the Assume that asymptotic expansion of the modified Bessel function of the . If is small first kind, we have and the log-likelihood ratio can be then . An equivalent decision rule is based approximated by on (12) is large then If ratio can be approximated by

and the log-likelihood (13)

, regardless of not Finally, note that in this case for low only the decorrelation operation on the input signal vanishes is zero. Thus we can see but also the bias term that the mixed-mode Ricean/Rayleigh optimal receiver does . not have any decorrelation features at low



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Fig. 2. Quadratic decorrelation receiver (QDR) for a mixed-mode Ricean/Rayleigh channel.

2) Quadratic Decorrelation Receiver (QDR): For extreme the decision variables (11)–(13) can be viewed values of and . Therefore, we as linear combinations of propose a suboptimal receiver based on the functional (14) where is a constant to be determined. For large values of , the constant should vanish as increases since in that . case the log-likelihood ratio can be approximated by . With The simplest function achieving this goal is tends to the true logthis, the decision variable likelihood ratio as assumes small or large values. Thus we obtain a family of receivers of the form (14) called Quadratic Decorrelation Receivers (QDR) whose decision variables are given by

signal-to-noise ratio (SNR) as well as at low SNR. It is to be noted that the QDR reduce to Aiken’s receiver [7] when the path magnitudes are Rayleigh-distributed. The QDR decision variable depends only on and . At low

thus we can see that the QDR although suboptimal also has the property that the decorrelation on the input signal vanishes at . low 3) Quadratic Receiver (QR): In order to assess the performance improvement due to the decorrelation operation, we must consider also receivers very similar to QDR except that they do not employ decorrelation. Therefore, we consider also simple Quadratic Receivers (QR) that are a limiting form of the QDR (15) when the multipath is resolved (i.e., when ). The decision variable for the QR is then

(15) and illustrated in Fig. 2. These receivers exploit the decorrelation performed on the input samples similar to the optimal receiver. However, the nonlinear term due to the specular component is replaced by a quadratic form that is more suitable for implementation. Furthermore, the QDR requires only. Calculation of the probability of error knowledge of for several values of showed that the best we could find is . This value gives a low probability of error at high

The QR can be considered as suboptimum receivers with respect to Turin’s resolved multipath optimum receiver [5]. When the path magnitudes are Rayleigh-distributed, they reduce to the optimum receiver for a resolved multipath


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Rayleigh fading channel. Calculation of the bit-error probabilities showed that for QR, the best we could find is . III. PERFORMANCE EVALUATION FOR BINARY SIGNALING Since the optimal receiver for Ricean multipath channels, (5), is not amenable to analysis, we will focus on the QDR. Therefore, these results will provide upper bounds for the error probability of the optimal receiver, and more important they will illustrate the performance of an implementable receiver. Furthermore, for high signal-to-noise ratio, the optimal receiver and the QDR converge to the same structure, showing that these upper bounds are tight. We will also investigate the performance of the QR scheme and compare with the QDR. This will show the improvement due to the decorrelation stage. Finally, by comparing the performance of the R OPT scheme with the QDR, we will assess the improvement yielded by knowledge of the Ricean specular term magnitude. With equiprobable equal energy binary signals and maximum-likelihood (ML) decision rule, the probability of error for the QDR with held fix is

(16) where

(i.e., ), the two resulting integrals can be evaluated using the residue theorem [25, p. 89]. two difficulties arise. First, the When probability of error is generally difficult to obtain precisely because of the complicated nature of the exponential factor . Second, in the characteristic function generally depends on the phase through the mean

thus in order to evaluate the probability of error, the pdf of needs to be integrated over and over the phases . of the form (17) may A characteristic function be diagonalized by use of the transformation

[26]. For sake of simplicity, the index will be omitted is to be chosen such that in the following. The matrix it diagonalizes the matrix while satisfying . is Hermitian, there exists an unitary matrix which Since (i.e., ). Let be the square root diagonalizes of ), exists since is positive definite. . Let denote the Form the Hermitian matrix , unitary matrix which diagonalizes the matrix , and . The matrix is given by . The and satisfy , , matrices and . The diagonalized characteristic function is given by

is a Hermitian quadratic form in jointly Gaussian random is a block-diagonal matrix with variables. The matrix diagonal blocks (18) and

and given by

. Using (3) and (4), the bias term

is

where . It is well known that (16) can be evaluated , of [24] by inverting the characteristic function,

. In this case, the residue method is where not practical since each residue consists of an infinite series. Therefore, when the multipath component magnitudes are Ricean-distributed, we will evaluate the probability of error by performing numerical inversion of the characteristic function. on the Now let us consider the dependency of . For a mixedphase . This dependency is via the mean mode Ricean/Rayleigh channel when only a single multipath is component is Ricean, it will be shown that independent of . It can be shown that the mean is given by

where and (4),

is given by (5f), and are

and are defined by (3) matrices defined by

When all ing to the

are null but one (for example, the one correspondth path, i.e., )

(17) and where probability density function (pdf) of , and transform of are obtained by integrating the pdf of

. The is given by the Fourier , . For Rayleigh channels



And

can be factored out in

. We have then

Since is independent of , the characteristic function is, in that special case only, independent of and from (18) it is given by (19) Therefore, in order to evaluate the probability of error for the mixed-mode channel it is not necessary to integrate over the phase, and the numerical inversion method proposed by Imhof in [27] can be used. Instead of the standard inversion formula

Imhof uses (20)

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can be bounded above by

The pair-wise probability of error evaluated by using the same method since

with

and

can be

.

IV. PERFORMANCE OF BINARY MODULATION SCHEMES OVER RAYLEIGH MULTIPATH CHANNELS The bit-error probabilities for the QDR and R QR schemes are presented as functions of the received signal-to-noise ratio . For a two-path channel, the received SNR is given per bit by

where and is the energy per bit of the real signal. In this paper we will consider commonly used modulation the schemes such as FSK and DPSK. Under complex envelope of a FSK binary signal is

denote the imaginary part of a function. Substitutwhere ing (19) into (20) and using the relations

and

the probability of error can be rewritten as (21)

where (carrier frequency) and is the symbol duration. FSK modulation with frequency separation will be denoted as FSK , where . For FSK, the observation interval is equal to the symbol duration plus the maximum of the channel interpath delays. Thus for FSK, , i.e., . we have Both conventional DPSK and symmetrical Differential PhaseShift Keying [28] (SDPSK) will be considered in this section. and the complex envelope of a DPSK binary Under signal over two symbol intervals is

where

and

We see that the problem of evaluating a line integral over a contour is reduced to evaluating an improper integral of a real function (21). Furthermore, it appears that the function is quite suitable for numerical integration. Moreover, is a removable singularity since the function the point at has a finite limit when . Since the function increases monotonically toward , the integration will be . The error of truncation carried over a finite range

Under and the complex envelope of SDPSK binary signal over two symbol intervals is


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Fig. 3.

Performance of FSK and DPSK signaling over a two-path Rayleigh fading channel.

Fig. 4.

Performance of FSK and DPSK signaling over a two-path Rayleigh fading channel.

Since in DPSK and SDPSK, the transition between the carrier phase of consecutive bits carries the information, the observation interval needs to be equal to twice the symbol duration. , Therefore, for DPSK and SDPSK, . The relative delay between the i.e., , is expressed as a first path and the second path, percentage of the symbol duration . The two-path Rayleigh fading channel is characterized by and the parameter , the

relative Rayleigh component strength between the first and the second path . Numerical results are presented in Figs. 3–6. Several parameter indexes pointing to the same curve show that corresponding curves overlap or are very close to each other. For clarity, only one of those curves is plotted. Figs. 3 and 4 present the performance of the QDR (equivalent to R OPT here) and R QR schemes over an unresolved multipath Rayleigh fading



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Fig. 5. Performance of FSK(1) and FSK(1=2) signaling over two- and three-path Rayleigh fading channels.

channel. Comparing the solid lines with the dot-dash lines, we can see that the performance degradation of the QDR due to unresolvability is between 3 and 8.5 dB in the error probability to . range of From Figs. 3 and 4 it is seen that the QDR scheme provides a twofold diversity-like gain even when the multipath is unresolved. Moreover, from Fig. 3 it is seen that for FSK and DPSK the R QR scheme has a marked error floor, while the QDR eliminates this effect. This shows the importance of the decorrelation operation on the input samples. Figs. 3 and 4 also show that the best performance is obtained ). However, from for equal path strength channels (i.e., Fig. 4 it is seen that even a low-power second path (for ) should not be discarded in the detection example, process because the QDR scheme may give some additional gain at high SNR compared to a simple noncoherent receiver, that does not exploit the second path. This is especially true , DPSK, and SDPSK. for FSK From Fig. 3 it is seen that DPSK and SDPSK detected with the QDR give the best performance. At high SNR, DPSK gives and between 10 and 4-dB improvement compared to FSK in the error probability range 13 dB compared to FSK to . However, 3 dB are gained because the observation interval used with DPSK is twice the one used with FSK. When the SNR is below 20 dB, FSK outperforms FSK but FSK gives better performance at higher SNR. Figs. 3 and 4 confirm that the performance of the R QR for small delay over two-path channels depends heavily on and SDPSK the R QR the modulation schemes. For FSK scheme performs reasonably well giving even diversity-like and DPSK however the R QR scheme gains. With FSK has error floors. These error floors are eliminated by the QDR. The improved performance of the QDR scheme can

be explained as follows. Over an equal path strength channel, the performance of the R QR scheme depends on two factors: 1) the two correlations matrices and and 2) the cross. These matrices are closely related to correlation matrix the shape of the signals and the interpath delay of the channel. For SDPSK and FSK, the correlation matrices are conjugate to each other having identical eigenvalues. On the other hand, for DPSK the two correlation matrices have different eigenvalues. Let us recall that the QDR scheme uses the eigenvalues of the , which are equal for SDPSK and matrices FSK and the R QR scheme uses the eigenvalues of the matrix . The relatively good performance of the R QR scheme for SDPSK can then be partly explained by the fact that though the R QR scheme does not use the appropriate eigenvalues, it uses an identical set of eigenvalues for both hypotheses similar has also to QDR. The form of the crosscorrelation matrix an important role since though the R QR scheme uses the as QDR, it yields some same set of eigenvalues for FSK error floor. The results for a two-path channel show that the performance of the QDR scheme is always better than the performance of the R QR scheme. It is to be expected that the gains and the superiority of the QDR scheme over R QR are even higher over a three-path channel as can be inferred from Figs. 5 and 6. With a notation similar to the two-path case, for a three-path channel, the received SNR is given by

where . The relative delay between the first path and the third path, , is expressed as a percentage of the symbol duration . The three-path Rayleigh fading channel


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Fig. 6.


Performance of DPSK signaling over two- and three-path Rayleigh fading channels.

is characterized by the value of the parameters (defined as which represents the Rayleigh for the two-path case) and path’s strength of the third path relative to the first. Figs. 5 and 6 show that the QDR scheme gives higher diversity-like gains over the three-path channel. For sufficiently high SNR, the QDR probability of error decreases by an order of magnitude , and thus yields diversity for an increase of 3.5 dB in gains of order three. Furthermore, from Fig. 5, it is seen that unlike over the two-path channel, the performance of R QR has an error floor over the three-path channel. This for FSK shows that the error floor phenomenon in the performance of the R QR scheme occurs more frequently over the three-path channel. However, this error floor is removed by the QDR. Therefore, we see that all the effects observed over the twopath channel are enhanced over the three-path channel. It is to be expected that the same trend exists as the number of paths is increased showing that the improvement of the QDR scheme might be even higher.

V. PERFORMANCE OF BINARY MODULATION SCHEMES OVER MIXED MODE RICEAN/RAYLEIGH CHANNELS In this section we present the performance of FSK , , DPSK, and SDPSK over two-path mixed-mode FSK Ricean/Rayleigh fading channels. Let us recall from Section II-D that in such a channel the first path gain is Riceanand the second path gain distributed with Ricean parameter is Rayleigh-distributed. This channel will then be characterized by the value of its parameters and K, where is defined in represents the Ricean parameter. Section IV and K For convenience K will be expressed in decibels. Similarly to Section IV, the bit-error probabilities will be presented as

functions of the received SNR per bit, which in this case is K

Numerical results are presented in Figs. 7–10. Since the influence of the parameter may be inferred from Section IV, we shall restrict our analysis to channels where the Rayleigh ). part of the path gains has equal strength (i.e., From Figs. 7–10 it is seen that the QDR scheme performs better than the R OPT scheme over mixed-mode Ricean/Rayleigh channels. Quantitatively, the QDR scheme and FSK gives up to 1-dB gain with respect with FSK for to the R OPT scheme at an error probability of 13 dB and at an error probability of for K 15 dB. K From Fig. 9 it is seen that the QDR scheme with DPSK gives up to 1-dB gain with respect to R OPT for K 13 dB in the and up to 1.5 dB for K error probability range of 15 dB at an error probability of . For SDPSK 0.7-dB gain is obtained at an error probability of for K 15 (see and 0.9 dB is obtained at an error probability of Fig. 10). This shows that SNR gains can be obtained by the use of receivers which exploit the knowledge of the specular component magnitude. Figs. 7–10 also show the superiority of the QDR scheme dB. over the QR at high SNR, i.e., when and Furthermore, Figs. 8 and 9 show that with FSK DPSK the QR scheme yields error floors which are eliminated by the QDR. At low SNR, the QR scheme performs the same or better than the QDR. However, the performance degradation of the QDR is in general small compared with the SNR gains which can be achieved by this scheme with respect to R OPT or QR at high SNR. For example, the QDR 13 dB at an error probability of yields 0.36-dB loss for K



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Fig. 7. Performance of FSK(1) signaling over a two-path Ricean/Rayleigh fading channel (K is expressed in decibels).

Fig. 8.

Performance of FSK(1=2) signaling over a two-path Ricean/Rayleigh fading channel (K is expressed in decibels).

for FSK and 0.3-dB loss for K 20 dB for . Note that when K DPSK at an error probability of 20 dB, the QR scheme with FSK outperforms the QDR scheme over the entire range of error probabilities considered in this paper. As shown in Section II-D, at low SNR the decorrelation vanishes for the mixed-mode Ricean/Rayleigh optimum receiver. This may explain why the QR outperforms

the QDR in some cases since the QR does not employ decorrelation. Nevertheless, considering the fact that the QDR eliminates the error floors, overall the superiority of the QDR over mixed-mode Ricean/Rayleigh channels is clear. Note that although it may appear on Figs. 7–10 that the QDR scheme gives asymptotically diversity-like gains of order higher than two, this is not actually the case. It has


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Fig. 9.


Performance of DPSK signaling over a two-path Ricean/Rayleigh fading channel (K is expressed in decibels).

Fig. 10. Performance of SDPSK signaling over a two-path Ricean/Rayleigh fading channel (K is expressed in decibels).

been verified that at lower probability of error the QDR over a mixed-mode Ricean/Rayleigh two-path channel behaves asymptotically as a twofold diversity system. VI. CONCLUSIONS In this paper we derived noncoherent detection structures for multipath Rayleigh/Ricean fading channels (optimum as

well as suboptimum). The multipath components are assumed to be unresolved, with known delays, yielding generalizations of the well-known RAKE receivers. We showed that the Rayleigh channel optimum receiver (R OPT) consist of an “orthogonalization” (or decorrelation) stage and then implements an optimum decision rule for a resolved multipath channel using the parameters of the transformed signal and channel.



We showed that the Ricean channel optimum receiver also performs this decorrelation in addition to nonlinear operations related to the specular term. These decorrelation operations vanish at low SNR. Lately, the use of a multipath decorrelation operation in code-division multiple-access (CDMA) systems has been mentioned in [29]. The context in which this operation is mentioned is an extension of the well-known decorrelation CDMA receivers. However, the relation of the multipath decorrelation stage to the optimal multipath receiver is not revealed in [29], nor its decisive role in an unresolved multipath channel. We show that with multipath decorrelation, diversity gains can be obtained also without spreading the signal bandwidth. Since the optimum decision rule over Ricean channels is in general too complex to implement, we proposed several suboptimum structures such as the quadratic decorrelation receiver (QDR) and the quadratic receiver (QR). The QDR scheme exploits the decorrelation performed on the input samples that vanishes for low SNR. The nonlinear term due to the Ricean specular component is replaced by a quadratic form that is more suitable for implementation. Single-pulse performance of the QDR, R OPT, and QR schemes were studied for commonly used binary modulation schemes such as variations of FSK and DPSK over two- and three-path Rayleigh channels and two-path mixed-mode Ricean/Rayleigh channels. For Rayleigh fading channels, the importance of using receivers especially designed to handle the unresolvability condition has been clearly demonstrated. Indeed, the results show that the QDR scheme (optimum in that case) is superior to the QR scheme (denoted R QR). The QDR eliminates the error floors, and it provides SNR gains with respect to the QR scheme. The QDR scheme gives diversity-like gains even if the channel is not fully tracked. It was shown that even a weak second path can contribute to diversity gains. For a two-path mixed-mode Ricean/Rayleigh channel, we showed that the additional knowledge of the power of the specular component can provide performance gains. Furthermore, we showed that in general the QDR scheme outperforms the QR also on mixed-mode Ricean/Rayleigh channels by eliminating the error floors. When K is larger than 20 dB, with FSK the QR scheme yields up to 1-dB gain compared to the QDR. In other situations, the QR may outperform the QDR by a fraction of a decibel; however, this advantage is shadowed by the error floor effect. Therefore, we maintain that overall the QDR scheme is a better choice. APPENDIX A EIGENVALUES AND EIGENFUNCTIONS OF THE SIGNAL PROCESS COVARIANCE , the eigenfunctions Under signal process are solutions of the integral equation

of the

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only the projection on the signal space is used in the decision rule. Therefore, the eigenfunctions associated with zero eigenvalues are irrelevant. Since is a finitedimensional kernel, the eigenfunctions associated with nonzero eigenvalues will be a linear combination of [22, p. 56]. For convenience, as (2) where we define the eigenfunctions are the coefficients of the linear combination, and . Substituting (2) into (22) and equating gives (omitting the index ) the coefficients of

Then the solution of (22) may be put into matrix form similar and are solution of the algebraic to Matthew [31], i.e., system (23) Moreover as shown in [22, p. 57], there is a complete equivalence between the integral (22) and the algebraic system (23) which is a classical eigenvalue problem. We see that the nonzero eigenvalues in the Karhunen–Loéve are the eigenvalues of the finite dimenexpansion for . Since and are both Hermitian sional matrix has real positive eigenvalues and positive definite, corresponding linearly independent eigenvectors [20, pp. denote the diagonal matrix with 230–232]. Let and let us define the matrix as , . then (23) is equivalent to Because of the equivalence of the integral equation and the algebraic system, the eigenfunctions will be uniquely of (23) which are exactly the determined by the solutions . Since is positive eigenvectors of the matrix definite, the eigenvalue problem (23) is equivalent to the generalized eigenvalue problem where . From [20, p. 231] we know that this linearly independent generalized eigenvalue problem has eigenvectors that can be chosen to be orthogonal (or . orthonormal) in the inner product defined by can be chosen such that Therefore, . Equivalently, can be chosen such that . In conclusion, the eigenvectors can be chosen to be orthonormal in the inner . It can be easily shown that this product defined by choice of eigenvectors yields orthonormal eigenfunctions. Furthermore, one can show that this is equivalent to . DERIVATION

APPENDIX B RECEIVER STRUCTURES

OF

A. Conditional Likelihood with (22) , the projection of the signal process on is If zero [30, p. 179]. Moreover, for detection in white noise,

be an -dimensional vector whose components Let on the eigenfunctions are the projections of associated with the covariance function of the signal process . Conditioned on , under each hypothesis, including , the signal is a complex Gaussian random process. Therefore,


2862


for each , forms a circularly complex Gaussian random vector [21, pp. 383–384]. From [30, p. 98] and [19] the conditional likelihood ratio given is found to be

where

and

Substituting the expressions for using (3) and (4) yields (6) and

and

into (26) and

where is given by (5d). Using scalar notation for purpose of integration, is also given by (5a).

are given by

B. Likelihood Ratios 1) -Path Ricean Channel: The likelihood ratios are obtained by integrating successively (25) using (5a) and with respect to all components of the between vector .

and

(24)

is given by (5f). From (24), we know that the where depends on , thus the conditional likelihood mean vector can be rewritten as (25) is the function that includes all factors involving is everything left over. For convenience, let us define

where and

(27) and are given by (5b) and (5c). where Using (3) and (4), (5d) yields (5e). We also define the vector as

Using (5e) and noting that get (7). 2) Two-Path Ricean Channel: Setting yields (after integration with respect to )

Note that

where is the real part of its argument, and and are “matrix operators” defined in Section II-A. Since is diagonal,

, from (6) we into (27)

(28) Let us define and

is independent of Hence

and given by

.

(26)

then and . Hence, we , , get the equality at the bottom of this page, where , , and . From and are, respectively, the phases of [23, p. 358] we have



where is the phase of of the phase of ) and yields (after integration with respect to )

(independent . Then (28)

It is seen that this likelihood ratio is independent of the phases in . ACKNOWLEDGMENT The first revision of this paper was completed while the authors were with IRC/Communications Lab of the Helsinki University of Technology, Espoo, Finland. The authors wish to thank the administration of the Communications Lab for the excellent conditions that were made available to complete the work. They would also like to thank the reviewers and the Editor for their useful comments. REFERENCES [1] “Special issue on Wireless Personal Communications,” IEEE J. Select. Areas Commun., vol. 11, Aug.–Sept. 1993. [2] “Special issue on Future PCS Technologies,” IEEE Trans. Veh. Technol., vol. 43, Aug. 1994. [3] “Special issue on Wireless Networks for Mobile and Personal Communications,” Proc. IEEE, vol. 82, Sept. 1994. [4] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, pp. 943–968, July 1993. [5] G. L. Turin, “Communication through noisy random-multipath channels,” in 1956 I.R.E. Nat. Conv. Rec., Mar. 1956, pt. 4, pp. 154–166. [6] R. Price and P. E. Green, “A communication technique for multipath channels,” Proc. IRE, vol. 46, pp. 555–570, Mar. 1958. [7] R. T. Aiken, “Communication over the discrete-path fading channel,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 346–347, Apr. 1967. , “Error probability for binary signaling through a multipath [8] channel,” Bell Syst. Tech. J., pp. 1601–1631, Sept. 1967.

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[9] J. E. Mazo, “Exact matched filter bound for two-beam Rayleigh fading,” IEEE Trans. Commun., vol. 39, pp. 1027–1030, July 1991. [10] M. Alles and S. Pasupathy, “Channel knowledge and optimal performance for two-wave Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 43, pp. 8–20, Feb. 1994. , “Suboptimum detection for the two-wave Rayleigh-fading chan[11] nel,” IEEE Trans. Commun., vol. 42, pp. 2947–2958, Nov. 1994. [12] R. Steele, Mobile Radio Communications. New York: IEEE Press, 1992. [13] T. Korhonen, M. Hall, and S.-G. Häggman, “Superresolution multipath channel parameter estimation by matched filter deconvolution and sequential bin tuning,” in 2nd Int. Work. Multi-Dimensional Mobile Communications (Seoul, Korea, July 1996). [14] Z. Kostic, M. I. Sezan, and E. L. Titlebaum, “Estimation of the parameters of a multipath channel using set-theoretic deconvolution,” IEEE Trans. Commun., vol. 40, pp. 1006–1011, June 1992. [15] J. Ehrenberg, T. Ewart, and R. Morris, “Signal processing techniques for resolving individual pulses in a multipath signal,” J. Acoust. Soc. Amer., vol. 63, no. 6, pp. 1861–1865, June 1978. [16] R. J. Figueiredo and A. Gerber, “Separation of superimposed signals by a cross-correlation method,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1084–1089, Oct. 1983. [17] J. P. Burg, “Maximum entropy spectral analysis,” in Proc. 37th Meet. Soc. Exploration Geophysicists, 1967. [18] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 276–280, Mar. 1986. [19] W. F. McGee, “Complex Gaussian noise moments,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 149–157, Mar. 1971. [20] J. M. Ortega, Matrix Theory. New York: Plenum, 1987. [21] H. V. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1988. [22] F. G. Tricomi, Integral Equations. New York: Interscience, 1957. [23] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1966. [24] G. L. Turin, “Some computations of error rates for selectively fading multipath channels,” in Proc. Nat. Electronics Conf., Mar. 1959, vol. 15, pp. 431–440. [25] M. A. Evgrafov, Analytic Functions. Philadelphia, PA: Saunders Mathematics Books, 1966. [26] G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika, pp. 199–201, June 1960. [27] J. P. Imhof, “Computing the distribution of quadratic forms in normal variables,” Biometrika, vol. 48, no. 3–4, pp. 419–426, 1961. [28] J. H. Winters, “Differential detection with intersymbol interference and frequency uncertainty,” IEEE Trans. Commun., vol. COM-32, pp. 25–33, Jan. 1984. [29] Z. Zvonar and D. Brady, “Linear multipath-decorrelating receivers for CDMA frequency-selective fading channels,” IEEE Trans. Commun., vol. 44, pp. 650–653, June 1996. [30] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968. [31] J. W. Matthews, “Eigenvalues and troposcatter multipath analysis,” IEEE J. Select. Areas Commun., vol. 10, pp. 497–505, Apr. 1992.


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